Chapter 8: Q 10 (page 704)

Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.
$\tan x, x_{0} = \frac{π}{3}$

Short Answer

Expert verified

The third-order Maclaurin series for the function $f (x) = \tan x$ is $P_{3} (x) = \sqrt{3} + (x - \frac{π}{3}) + \frac{1}{3} {(x - \frac{π}{3})}^{3}$

Step by step solution

Given information

The function is $f (x) = \tan x$

The third Maclaurin polynomial is given for any function f with a derivative of order 3 at x=0

The third-order Taylor polynomial at $x_{0} = \frac{π}{3}$ is

$P_{3} (x) = f (\frac{π}{3}) + f^{'} (\frac{π}{3}) (x - \frac{π}{3}) + \frac{f^{''} (\frac{π}{3})}{2!} {(x - \frac{π}{3})}^{2} + \frac{f^{''} (\frac{π}{3})}{3!} {(x - \frac{π}{3})}^{3}$

First, determine the function's value as well as $f^{'} (x), f^{''} (x) a n d f^{'''} (x) a t x_{0} = \frac{π}{3}$

Find the derivatives of the function

The value of the function $x = \frac{π}{3}$ is

$f (\frac{π}{3}) = \tan (\frac{π}{3}) = \sqrt{3}$

The derivatives of the function $f (x) = \tan x$ are

$f^{'} (x) = \frac{d}{d x} [\tan x] = s e c^{2} x$

At $x = 0$

$f^{'} (0) = s e c^{2} 0 = 1$

Again

$f^{''} (x) = \frac{d}{d x} (s e c^{2} x) = 2 s e c x \cdot s e c x \tan x = 2 s e c^{2} x \tan x$

At $x = 0$

$f^{''} (0) = 2 s e c^{2} 0 \tan 0 = 2 \cdot 1 \cdot 0 = 0$

Again

$f^{''} (x) = 2 \frac{d}{d x} [\sec^{2} x \tan x] = 2 [\sec^{2} x \frac{d}{d x} (\tan x) + \tan x \frac{d}{d x} (\sec^{2} x)] = 2 [\sec^{2} x \cdot \sec^{2} x + \tan x \cdot 2 \sec x \cdot \sec x \tan x] = 2 [\sec^{4} x + 2 \sec^{2} x \tan^{2} x]$

At $x = 0$

$f^{'''} (0) = 2 (\sec^{4} 0 + 2 \sec^{2} 0 \tan^{2} 0) = 2 (1 + 2 \cdot 1 \cdot 0) = 2$

Find the third-order Maclaurin series for the function.

The third-order Maclaurin series for the function $f (x) = \tan x$ at $x = \frac{π}{3}$ is

$P_{3} (x) = \sqrt{3} + 1 \cdot (x - \frac{π}{3}) + \frac{0}{2!} {(x - \frac{π}{3})}^{2} + \frac{2}{3!} {(x - \frac{π}{3})}^{3} P_{3} (x) = \sqrt{3} + (x - \frac{π}{3}) + \frac{1}{3} {(x - \frac{π}{3})}^{3}$

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Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.
$\tan x, x_{0} = \frac{π}{3}$

Short Answer

Step by step solution

Given information

The third Maclaurin polynomial is given for any function f with a derivative of order 3 at x=0

Find the derivatives of the function

Find the third-order Maclaurin series for the function.

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Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.tanx,x0=π3

Short Answer

Step by step solution

Given information

The third Maclaurin polynomial is given for any function f with a derivative of order 3 at x=0

Find the derivatives of the function

Find the third-order Maclaurin series for the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Discrete Mathematics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.
$\tan x, x_{0} = \frac{π}{3}$